Michaelis population kinetics

The structural submodel describes the central tendency of the time course of the antibody concentrations as a function of the estimated typical pharmacokinetic parameters and independent variables such as the dosing regimen and time. As described in Section 3.9.3, mAbs exhibit several parallel elimination pathways. A population structural submodel to mechanistically cover these aspects is depicted schematically in Fig. 3.14. The principal element in this more sophisticated model is the incorporation of a second elimination pathway as a nonlinear process (Michaelis-Menten kinetics) into the structural model with the additional parameters Vmax, the maximum elimination rate, and km, the concentration at which the elimination rate is 50% of the maximum value. The addition of this second nonlinear elimination process from the peripheral compartment to the linear clearance process usually significantly improves the fit of the model to the data. Total clearance is the sum of both clearance parts. The dependence of total clearance on mAb concentrations is illustrated in Fig. 3.15, using population estimates of the linear (CLl) and nonlinear clearance (CLnl) components. At low concentra-... 

FIGURE 3.11. Change of reaction rate (A) with a rate factor according to the Michaelis-Menten kinetics and (B) with time and population response. 

The Michaelis-Menten kinetics model, illustrated for a lake in Example 2.20, may also be applied to a flowing stream in which the microorganisms are attached to the surfaces of the charmel, have a relatively steady cell density, and are exposed to the full chemical concentration in the stream (Cohen et al., 1995 Kim et al., 1995). Microorganisms attached to solid surfaces form biofilms, as populations of attached microbes accumulate on top of one another, building up a layer of microbes embedded in an extracellular matrix which they secrete. Within biofilms, the microbial cell density X corresponds to the number of attached microorganisms divided by the volume of the biofilm. In wastewater treatment engineering, a biofilm is often referred to as attached growth. A biofilm may also be called a bacterial... 

Microbial Biotransformation. Microbial population growth and substrate utilization can be described via Monod s (35) analogy with Michaelis-Menten enzyme kinetics (36). The growth of a microbial population in an unlimiting environment is described by dN/dt = u N, where u is called the "specific growth rate and N is microbial biomass or population size. The Monod equation modifies this by recognizing that consumption of resources in a finite environment must at some point curtail the rate of increase (dN/dt) of the population ... 

Now we consider situations in which transformation of the organic compound of interest does not cause growth of the microbial population. This may apply in many engineered laboratory and field situations (e.g., Semprini, 1997 Kim and Hao, 1999 Rittmann and McCarty, 2001). The rate of chemical removal in such cases may be controlled by the speed with which an enzyme catalyzes the chemical s structural change (e.g., steps 2, 3 and 4 in Fig. 17.1). This situation has been referred to as co-metabolism, when the relevant enzyme, intended to catalyze transformations of natural substances, also catalyzes the degradation of xenobiotic compounds due to its imperfect substrate specificity (Horvath, 1972 Alexander, 1981). Although the term, co-metabolism, may be used too broadly (Wackett, 1996), in this section we only consider instances in which enzyme-compound interactions limit the overall substrate s removal. Since enzyme-mediated kinetics were characterized long ago by Michaelis and Menten (Nelson and Cox, 2000), we will refer to such situations as Michaelis-Menten cases. 

Now we can see the types of biochemical factors that determine the rate constant, fcbio for Michaelis-Menten cases the ability of the enzyme to catalyze the transformation as reflected by the quotient, kE/KiMM, and the presence of enzyme in the microorganism population involved, as quantified by [Enz]tot/[B], In the following section, we develop some detailed kinetic expressions for one case of enzyme-mediated transformations. Examination of these results will help us to see how structural features of xenobiotic compounds may affect rates. Finally, we will improve our ability to understand the relative rates for structurally related chemicals that are transformed by the same mechanism and are limited at the same biodegradation step. 

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